2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda-delta/substitution/leq.ma".
13 include "lambda-delta/substitution/lift.ma".
15 (* DROPPING *****************************************************************)
17 inductive drop: lenv → nat → nat → lenv → Prop ≝
18 | drop_sort: ∀d,e. drop (⋆) d e (⋆)
19 | drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
20 | drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
21 | drop_skip: ∀L1,L2,I,V1,V2,d,e.
22 drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
23 drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
26 interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
28 (* Basic inversion lemmas ***************************************************)
30 lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
31 #d #e #L1 #L2 #H elim H -H d e L1 L2
33 | #L1 #L2 #I #V #_ #IHL12 #H1 #H2
34 >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
35 | #L1 #L2 #I #V #e #_ #_ #_ #H
36 elim (plus_S_eq_O_false … H)
37 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
38 elim (plus_S_eq_O_false … H)
42 lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
45 lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
47 #d #e #L1 #L2 * -d e L1 L2
49 | #L1 #L2 #I #V #_ #H destruct
50 | #L1 #L2 #I #V #e #_ #H destruct
51 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
55 lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
58 lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
59 ∀K,I,V. L1 = K. 𝕓{I} V →
60 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
61 (0 < e ∧ ↓[d, e - 1] K ≡ L2).
62 #d #e #L1 #L2 * -d e L1 L2
63 [ #d #e #_ #K #I #V #H destruct
64 | #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
65 >(drop_inv_refl … HL12) -HL12 K /3/
66 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
67 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
71 lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
72 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
73 (0 < e ∧ ↓[0, e - 1] K ≡ L2).
76 lemma drop_inv_drop1: ∀e,K,I,V,L2.
77 ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
78 #e #K #I #V #L2 #H #He
79 elim (drop_inv_O1 … H) -H * // #H destruct -e;
80 elim (lt_refl_false … He)
83 lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
84 ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
85 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
88 #d #e #L1 #L2 * -d e L1 L2
89 [ #d #e #_ #I #K #V #H destruct
90 | #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
91 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
92 | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
97 lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
98 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
103 lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
104 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
105 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
106 ↑[d - 1, e] V2 ≡ V1 &
108 #d #e #L1 #L2 * -d e L1 L2
109 [ #d #e #_ #I #K #V #H destruct
110 | #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
111 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
112 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
117 lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
118 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
122 (* Basic properties *********************************************************)
124 lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
128 lemma drop_drop_lt: ∀L1,L2,I,V,e.
129 ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
130 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
133 lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
134 ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
136 ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 &
137 ↓[0, i] L2 ≡ K2. 𝕓{I} V.
138 #L1 #L2 #d #e #H elim H -H L1 L2 d e
139 [ #d #e #I #K1 #V #i #H
140 lapply (drop_inv_sort1 … H) -H #H destruct
141 | #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
142 elim (lt_zero_false … H)
143 | #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
144 elim (drop_inv_O1 … H) -H * #Hi #HLK1
145 [ -IHL12 Hie; destruct -i K1 J W;
146 <minus_n_O <minus_plus_m_m /2/
148 elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
150 | #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
151 lapply (plus_S_le_to_pos … Hdi) #Hi
152 lapply (drop_inv_drop1 … H ?) -H // #HLK1
153 elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
157 (* Basic forvard lemmas *****************************************************)
159 lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
162 [ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
163 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
164 elim (drop_inv_O1 … H) -H * #He #H
165 [ -IHL1; destruct -e K2 I2 V2 /2/
166 | @drop_drop >(plus_minus_m_m e 1) /2/
171 lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
172 ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
174 [ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
175 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
176 elim (drop_inv_O1 … H) -H * #He #H
177 [ -IHL1; destruct -e K2 I2 V2 //
178 | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
183 lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
185 [ #L2 #e #H >(drop_inv_sort1 … H) -H //
186 | #K1 #I1 #V1 #IHL1 #L2 #e #H
187 elim (drop_inv_O1 … H) -H * #He #H
188 [ -IHL1; destruct -e L2 //
189 | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
190 >minus_le_minus_minus_comm //